--- sort: 070 title: Curves and Surfaces flashcard: "Orals::Curves and Surfaces" --- # Curves and Surfaces - [x] What is **Zariski's main theorem**? #syllabus ✅ 2022-12-06 - [ ] For $f:X\birational Y$ a birational projective morphism of Noetherian integral schemes with $Y$ normal, every fiber $f\inv(y)$ is connected. - [ ] For $T:X\to Y$ a birational morphism of projective varieties with $X$ normal and $p$ a *fundamental point* of $T$, then the total transform $T(p)$ is connected of dimension $d\geq 1$. ($T$ is defined on some largest open $U\subseteq X$, and $X\sm U$ are the fundamental points. Here the total transform is $T(p) \da p_2 p_1\inv(p)$ where $\Gamma_\phi \subset U \cross Y \mapsvia{p_1, p_2} U, Y$ is the graph of a representative $\phi: U\to Y$.) - [x] Discuss Zariski's main theorem. #syllabus ✅ 2022-12-06 - [ ] Used to factor birational transformations of surfaces: if $f:X' \to X$ is a birational transformation of smooth projective surfaces and $p$ is a fundamental point of $f\inv$, then $f$ factors through the blowup $\pi: \Bl_p X \to X$. So any birational transformation $T: X_1\to X_2$ can be factored as a finite sequence of blowups and blowdowns. - [x] Need a surface have finitely many **exceptional curves of the first kind**? ✅ 2022-12-06 - [ ] No: $\Bl_n \PP^2$ for $n\geq 9$ has infinitely many. - [x] Discuss **classification of surfaces**. ✅ 2022-12-06 - [ ] Define $\kappa(X) \da [\bigoplus_{n\geq 0} H^0(\OO_X(nK_X)):\, k]_\trdeg$. - [ ] $\kappa = -1$: Rational or ruled surfaces - [ ] $\kappa = 0$: K3, Enriques, AV, or hyperelliptic surfaces - [ ] $\kappa=1$: Elliptic surfaces - [ ] $\kappa=2$: Surfaces of general type - [x] Show that a line bundle is globally generated iff basepoint free. ✅ 2022-12-01 - [ ] Globally generated: $\exists \OO_X\sumpower I \surjects L$ a surjection of sheaves; bpf means $\forall x\in X \exists s\in H^0(L)$ such that $s_x\not\in \mfm_x \subseteq \OO_{X, x}$, so $s_x$ is a unit and $L_x\cong \OO_{X, x}$ is generated by $s_x$ as an $\OO_{X, x}\dash$module. If $L$ is bpf, for each $x_i\in X$ choose $s_i\in H^0(L)$ so the image $s(x_i) \in L_{x_i}$ is nonzero and assemble $\ts{s_i}$ into a map $\OO_{X}\sumpower{\abs{X}} \to L$ -- this is surjective on stalks, since the fibers are 1-dimensional and thus generated by any nonzero element, and thus surjective. - [x] Give an example of globally generated $\neq$ basepoint free for general sheaves. ✅ 2022-12-01 - [ ] Take any $\mcf$ which is not globally generated and set $L = \OO_X \oplus \mcf$. Then $L$ is bpf since $\OO_X$ is but not globally generated since $\mcf$ is not. Similarly the zero sheaf is globally generated but not bpf. - [x] What is a **hyperelliptic curve**? ✅ 2022-12-07 - [ ] $g\geq 2$ and admits a finite degree 2 morphism $f:X\to \PP^1$. - [x] What is the ideal sheaf for $C\embeds S$ a curve in a surface? What is $I/I^2$? ✅ 2022-12-06 - [ ] $\mci_C = \OO_S(-C)$ and $\mci_C/\mci_C^2 = \OO_S(-C)\tensor \OO_C$. - [x] What is the canonical for $\PP^2$? What is its self-intersection? ✅ 2022-12-07 - [ ] $K = -3H$ so $K^2 = 9$. - [x] When is $K_C$ very ample? ✅ 2022-12-06 - [ ] For $g\geq 2$, $K_C$ is very ample iff $C$ is not hyperlliptic (admits a finite morphism $C\to \PP^2$ of degree 2). - [x] When is $K_C$ BPF? ✅ 2022-12-06 - [ ] For any $C$ with $g(C) \geq 2$. True because $\dim\abs{K-p} = \dim \abs{K} - 1$, using that $\dim \abs{K} = h^0(\omega_C)-1 = g-1$, $\dim \abs{p} = 0$ since $C$ is not rational, and apply RR to get $\dim \abs{K-p}=g-2$. - [x] What is the **canonical embedding**? What is its degree? ✅ 2022-12-07 - [ ] For $C$ hyperelliptic with $g\geq 3$, the embedding determined by $K_C$ which is very ample. Is degree $2g-2$. - [x] What is the **Riemann-Roch** theorem for curves? ✅ 2022-12-01 - [ ] $\chi(\OO_C(D)) = \deg D + (1-g)$. How to remember: $D= 0$ yields $h^0(\OO) - h^1(\OO) = 1-g$ and $h^1(\OO) = h^{0, 1} = h^{1,0} = h^0(\Omega) = p_g(C)$, the geometric genus, and $H^0(\OO) = k$ for a proper curve. More generally, $h^0(D) - h^0(K-D) = \deg(D) + (1-g)$. - [x] How is RR proved? ✅ 2022-12-01 - [ ] Show for $D=0$, then show it holds for $D$ iff for $D+p$ for $p\in C$ any point. - [ ] For $D=0$: the formula is $\chi(\OO_X) = 0+1-g$ which is true since the LHS is $h^0(\OO_X) = \dim_k k = 1, h^1(\OO_X) = g$. - [ ] For $D+P$: take $\mcl(-p)\injects \OO_X \surjects \iota^p_* k$ where $\iota^p: \ts{p}\injects C$ so the last term is the skyscraper sheaf for $k$ at $p$. - [ ] Twist by $\mcl(D+p)$ to get $\mcl(D) \injects \mcl(D+p) \surjects \iota^p_* k$ and $\chi(\mcl(D+p)) = \chi(\mcl(D)) + \chi(\iota^p_* k) = \chi(\mcl(D)) + 1$ and conclude using $\deg(D+p) = \deg D + 1$. - [x] What is the **Riemann-Hurwitz** theorem for curves? ✅ 2022-12-01 - [ ] $$\chi(X) = \deg(f)\cdot\chi(Y) + \deg R,\qquad \deg R = \sum_{p\in X} (e_p-1)$$ - [x] What is the **ramification index** $e_p$? ✅ 2022-12-01 - [ ] For a finite morphism $f:X\to Y$ of curves and $p\in X, q=f(p)\in Y$, let $t\in \OO_{f(x), Y}$ be a local parameter, push to $\OO_{p X}$ via $f^\sharp: \OO_{f(x), Y} \to \OO_{p, X}$ and define $e_p \da v_p(t)$ where $v_p$ is the valuation in the DVR $\OO_{p, X}$. If $e_p > 1$ then $f$ is ramified at $p$ and $f(p)$ is a branch point; if $e_p = 1$ then $f$ is unramified at $p$. - [x] What is **tame/wild ramification**? ✅ 2022-12-01 - [ ] For $\characteristic k = 0$ or $p$ where $p\not\mid e_p$, **tame**. Otherwise if $\characteristic k = 0$ with $p\mid e_p$, **wild**. - [x] Why is Riemann-Hurwitz true? ✅ 2022-12-01 - [ ] Take degrees in $K_X \sim f^* K_Y + R$ where $R \da \sum_{p\in X} \length(\Omega_{X/Y})_p\cdot p$ is the ramification divisor. - [x] What is $\deg K_C$ for $C$ a curve? ✅ 2022-12-06 - [ ] Set $D=K$ to get $h^0(K) - h^0(0) = \deg K + 1-g$, note $h^0(K) = p_g(C) = g$ and $h^0(0) = 1$ to get $g-1 = \deg K + 1-g \implies \deg K = 2g-2$. - [x] What is a **special/nonspecial divisor**? ✅ 2022-12-06 - [ ] $D$ Special: $h^0(K-D) > 0$, nonspecial otherwise. - [x] Give an example of a nonspecial divisor. ✅ 2022-12-06 - [ ] $D$ with $\deg D \geq 2g-2$, then $\deg (K-D) < 0 \implies h^0(K-D) = 0$. - [x] What is $\deg K_C$ for $C$ an elliptic curve? ✅ 2022-12-06 - [ ] $\deg K_C = 2g-2 = 0$. - [x] How is the canonical divisor changed under a morphism of curves? ✅ 2022-12-06 - [ ] For $f: X\to Y$ finite separable, $K_X \sim f^* K_Y + R$ the ramification divisor. - [x] When is $D\in \Div(C)$ **basepoint free**? #syllabus ✅ 2022-12-06 - [ ] The complete linear system $\abs{D}$ is BPF iff $\forall p\in C$, $\dim\abs{D-p} = \dim\abs{D}-1$, so the dimension drops by exactly one. - [ ] Holds if $\deg D \geq 2g$: check that $h^0(K-D) = h^0(K-(D-p)) = 0$ since $\deg(K-D) \leq 2g-2-2g \leq -2 < 0$ and $\deg(K-(D-p)) \leq 2g-2-2g+1 \leq -1 < 0$. Apply RR twice, first to $D$ and then to $D-p$: $$h^0(D) + h^0(K-D) = \deg(D) + 1-g \implies \dim \abs{D} + 0 = \deg(D) + 1-g$$ so $\dim\abs{D} = \deg(D) + 1-g$, and $$h^0(D-p) + h^0(K-(D-p)) = \deg(D-p) + 1-g = \deg(D) -1 + 1 - g = \deg(D) -g$$ so $$\begin{align*}\dim\abs{D-p} &= \deg(D) - g \\ \implies \deg(D) &= \dim\abs{D-p} + g \\ \implies \dim \abs{D} &= \qty{\dim \abs{D-p} + g} + 1-g \\ \implies \dim \abs{D} &= \dim \abs{D-p}+1\end{align*}$$ - [x] When is $D\in\Div(C)$ **very ample**? #syllabus ✅ 2022-12-06 - [ ] Iff $\forall p, q\in C$ (including $p=q$), $\dim \abs{D-p-q} = \abs{D} - 2$. - [ ] Holds if $\deg D \geq 2g+1$: Check that $D$ and $D-p-q$ are nonspecial since $$\deg(K-D) \leq 2g-2-(2g+1) \leq -3$$ and $$\deg(K-(D-p-q)) \leq 2g-2-(2g+1-1-1) \leq -1,$$so RR gives $$\begin{align*}h^0(D) &= \deg(D) + 1-g \\h^0(D-p-q) &= \deg(D-p-q) + 1-g = \deg(D) -1-g \\ \\\implies h^0(D) - h^0(D-p-q) &=(1-g) - (-1-g) = 2\end{align*}$$ - [x] When is $D\in \Div(C)$ **ample**? #syllabus ✅ 2022-12-06 - [ ] Iff $\deg (D) > 0$. $\implies$: $D$ ample $\implies nD$ very ample $\implies nD\sim H$ for $H$ a hyperplane section of the embedding and $\deg H > 0\implies \deg(D) > 0$. $\impliedby$: $\deg(D) > 0\implies nD \geq 2g+1$ for $n\gg 0$ $\implies nD$ is very ample (by the RR criterion). - [x] Discuss ampleness of divisors in $\PP^1$. ✅ 2022-12-06 - [ ] $\deg D \geq 1 \iff$ample $\iff$ very ample. - [x] If $D$ is a very ample divisor, discuss $\deg \phi$ for $\phi: C\to \PP^N$. ✅ 2022-12-06 - [ ] $\deg \phi = \deg D$. - [x] What is the **classification of curves in ${\mathbb{P}}^n$**? ✅ 2022-12-06 - [ ] There is a point $O\in \PP^3\sm X$ such that projection away from $O$ gives a birational morphism $X\birational X'\subseteq \PP^2$ where $X'$ is at worst nodal. Thus any curve (at all) is birational to a nodal curve. - [x] What is the **adjunction formula** for smooth curves on surfaces? #syllabus ✅ 2022-12-06 - [ ] $$2g-2 = C.(C+K_X).$$ - [x] What is the **adjunction formula** for divisors? ✅ 2022-12-11 - [ ] For $X$ smooth projective and $D$ a smooth prime divisor, $$K_D = (K_X +D)\mid_D.$$ - [x] How is the **degree of curve** on a surface defined? ✅ 2022-12-06 - [ ] Fix an ample divisor $H$, then $C.H$ is the degree of $C$ in the projective embedding determined by $H$, so define $\deg C = C.H$. - [x] When is a divisor on a surface **ample**? ✅ 2022-12-06 - [ ] $D$ is ample iff $D^2 > 0$ and $D.C > 0$ for all irreducible curves in $X$. - [x] Discuss formulas for blowups. ✅ 2022-12-06 - [ ] For $\tilde X \mapsvia{\pi} X$, $C.D = (\pi^* C).(\pi^* D)$, $(\pi^* C). E= 0$, $E^2 = -1$, and $(\pi^* C).D = C.(\pi_* D)$. For canonicals, $K_{\tilde X}^2 = K_X^2 - 1$ and $K_{\tilde X} = \pi^* K_X + E$, and $\Pic \tilde X = \gens{E}_\ZZ \oplus \Pic X$. - [x] What is the **strict transform**? ✅ 2022-12-06 - [ ] For $\tilde X\to X$ a blowup at $p$ and $C\subset X$, the closure $\tilde C$ in $\tilde X$ of $\pi\inv(C\intersect (X\sm p))$. Informally: $\tilde C = \pi^* C \sm E$. - [x] What is the **genus-degree formula** for curves in $\PP^2$? ✅ 2022-12-02 - [ ] $$g = {1\over 2}(d-1)(d-2)$$ Prove by adjunction: $2g-2 = C.(C+K_{\PP^2}) = C.(C-3H) = d(d-3)$ and rearrange.